**Early Operation Tools on Arithmetic’s System Section 01**

Measurements and calculations that have an operational tool in which will perform to and form of any kind of groups numbers, like rational numbers, original numbers, decimal numbers, etc. which will be called as arithmetic’s system. There are two fundamental standard operational tools in calculation and measurements, which are, addition, and subtraction. With several forms of a derived standard operational tool like multiplication, division, involutions, and/or evolution.

**Addition and/or Summation**

Addition showed as a collective group of numbers to become one. In the very early generation, if a caveman would like to know about how many animals have they capture if they adding the first two animals they capture, to the later three animals that captured, the caveman will lay down on the ground the last three animals that been capture, also they will lay down the first two animals they capture earlier, then they will counting on how many animals in total they have captured, without an idea of unit measurement yet. After trial and error in several times application of this situation, then the caveman will not be laying down on the ground anymore for counting on how many animals they have captured. Now, the caveman will know, if the caveman has capture two animals in the morning, and three animals in the noon, then by the end of the day the caveman will always have a summation of five animals in total, therefore with this experiments that build an initiate fundamental operation since the early existence of human being by adding three with two will always giving a summation of five regarding their unit measurements. With that situation, this also will build to more trials and errors till the concept of addition got built naturally in their concept of existence in thinking to perform more variant of addition format like if the caveman has one capture animal in the morning and one capture animal in the afternoon, then the caveman will have two capture animals by the end of the days, or if the caveman has one capture animal in the morning and another two capture animals in the afternoon, then the caveman will have three capture animals by the end of the day, and so on.

By the early system of addition, most of the human existence at that time will be able to be counting whatever as long as the person can adding it on single digits form, like nine and nine, 9+9. However, if those situations got risen from a single digit to a form of dual digits addition, like twenty-one and thirty-six, 21+36. With notes, about twenty-one is two tens of unit with one of a single unit, meanwhile, thirty-six is three tens of unit with six of a single unit. We can be counting those kinds of the situation by a system of additions like:

Two tens of unit + one of single unit => 2 tens + 1 unit

Three tens of unit + six of single unit => 3 tens + 6 unit

With the idea in the first steps, we are adding the single unit, which is one of a single unit and six of a single unit to perform a summation of seven of a single unit, 1 + 6 = 7. As we see it, once we know about the summation of two single units doesn’t perform a dual digit, then we can just continue it by counting that seven of single units as a summation part of single units of twenty-one and thirty-six. Now, we can continue with the tens of units, which in this case is two tens of units and three tens of units, the combination of these two will perform a summation of five tens of units, 20 + 30 = 50. Therefore, with that tens of unit calculation in this case, now we will have a final summation in the format as five tens of units and seven of a single unit, which is once we know it, this five tens of units and seven of a single unit can be performed numerically in a format of numbering as 57.

On the other hand, if we have situations by mean like the summation from adding two single-digit of the single unit becoming two digits of single units, then we will need to know this new form of two digits like a format of the tens of unit or the other in the "initiate form". Like in our example above, the twenty-one, in numbering is 21, which means 2 of tens of unit, and 1 of a single unit, the format of 2 of tens of unit in a group of 21 is the initiate numbering form. Similarly, with the format on a group of numbering in 36, which mean the 3 of tens of unit in that group is the "initiate form". Harmoniously in a case like in the format of hundreds of units, like 365, which mean three of hundreds of unit with six of tens of unit and five of a single unit, in numbering form like 300 + 60 + 5, then the initiate numbering form will be 3, instead of 6, because the three placed earlier in 365 numbers.

With that initiate numbering form mind, if our addition form has such a form like twenty-seven and thirty-five, then we know it now about twenty-seven is the form of two of tens of unit with seven of the single unit, like 20 + 7, meanwhile, thirty-five is the form of three of tens of unit with five of the single unit, like 30 + 5. With those two-addition forms, we will get a summation as sixty-two, 62, because the addition of two single units in this situation is seven, and five will give us twelve, 12, which is one of the tens of unit with two of a single unit, 10 + 2. We place the two of a single unit as the final result in the summation for single unit section, then we can proceed to the summation of tens of unit in this format, which is originated of two of tens of the unit and three of tens of the unit, with an addition that comes from the combination of a single unit, which is one of tens oof unit, therefore, once we combine all of these tens of units, 20 + 30 + 10, we will have six of tens of units, which called as 60, as our section in the tens of unit. In conclusion, with the end of the final summation, then we will have six of tens of unit with two of a single unit, which is 60 + 2, and that perform as sixty-two, 62, and that will be our final result in our form of summation for two additions between 27 + 35.

Written like: `27+35 = 62`

**Subtraction and/or Difference**

In the form of subtraction, we will be trying to take out one or several objects from a number of objects that exist. Let’s say if our caveman having nine captured animals and want to be finding out how many captured animals left once the caveman makes barbeque meat from three captured animal for dinner. Then, as usual for the caveman in the early counting method on operations, the caveman will lay down on the ground those nine captured animals, then take three captured animals out from the ground, and start counting on how many captured animal left on the ground, which the caveman will know about the subtraction from nine captured animal to three captured animal will give difference as six of captured animal left. With that in mind, the caveman will know now about if there’s a missing captured animal because of any reason, like become barbeque meat or being lost in the way, then there will always be a difference from the total. The caveman will also able to perform this fundamental calculation in their mind once they have enough trial and error about these differences, also they will try to perform this subtraction system in some other form, an example like if the first group have nine capture animals and they give five of capture animals to other hunting groups, then the first group will only have four of captured animals left. Also if the first group have nine capture animals and some group stealing two capture animals which make it only seven captured animal left. These fundamental operation tools all the caveman needed in order of counting about their stocks on their supply to survive and/or to feeding their family to support the daily activity, these additions to finding the summation and this subtraction to finding the difference, have helped more than centuries on many activities in order to do counting of some objects and/or materials around them.

**Multiplication and/or Product**

Multiplication is the other form of additions. If we want to know about how to be counting the multiplication, we will be able to find an answer to questions of five times seven by doing a simple group of additions, because five times seven means there are five additional of seven, such that 7 + 7 + 7 + 7 + 7, or there are seven additional of five, such that 5 + 5 + 5 + 5 + 5 + 5 + 5. By the concept of additional, that caveman may have left the legacy through some possible way from generation to generations, till we discover this multiplication system, we know it now about by adding five of those sevens or sevens of those five, then we will have thirty-five as our final summation. By the times with the trials and errors, a group of peoples in the early era operational tools discover about for each similar condition by adding one similar number several times will be giving such a special result, which in our first example by adding five of those sevens, 7 + 7 + 7 + 7 + 7, or seven of those five, 5 + 5 + 5 + 5 + 5 + 5 + 5, then we will always get thirty-five. By gradually developing and discovering in process of an operational tool on counting, we got perform tons of list of similarity existences in this repetition of counting by some variant of a single group of numbers, like three of eights can be performed as three times eights, in form of 8 + 8 + 8, which also introduce a new symbol of multiplication written as 3 x 8, which later become 3 * 8, or eights of three can be performed as eight times three, in form of 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3, which also can be written as 8 * 3. Or seventh of nine, which means seven times nine, in form of 9 + 9 + 9 + 9 + 9 + 9 + 9, which also can be simplify written as 7 * 9. This model of multiplication, later on, gets developed to a format of products of twelve. But, in short, once we truly can understand instead of knowing only for the idea of multiplication till the ninth of nine, in terms of 9 * 9, then we will be able to do the counting in multiplication by whichever kind of format.

In the meantime, of course, that multiplication can be identified by memorizing those groups if we are good enough at memorizing it, or we could also understand it through some pattern that exists among those groups of the number itself. By the growing of community in human knowledge on learning and interest, then several people usually modifying this method of multiplication by some of the variant method, which one of them called as *duplation*, which is a counting system that based on the counting of two and addition. An example, if we want to perform on finding the product of two factors like 24 and 18, format 24 * 18. We could perform a counting by terms like:

a | b | c | d |

1 | 24 | ||

2 | 48 | 2 | 48 |

4 | 96 | ||

8 | 192 | ||

16 | 384 | 16 | 384 |

432 |

As we have seen, the form of 432, in the last row and last column is the final answer on our calculation. This performance may look difficult, even though in practice this is super easily simple.

As much as we know, we perform 4 columns, (a), (b), (c), and (d). the first digit in column (a) will be 1; the first digit in column (b) will be a number that we want to multiply, which in this case is 24. A number that will be multiplied is a number that needed to get multiplied by some other number, which is often called a multiplier number. We will keep multiplying the number on column (a) by a factor of 2 till we can reach a number that closed to the equals or almost reach the multiplier, which is 18. We stop by 16 because 2 multiplied by 16 is 32, which is much more estimable than a group of number 18.

Now, once we doubling the numbers on column (b), which will be giving us a result of 48. Then we will double the 48 and keep continuing the doubling in column (b) till we reach the value of a number that reaching numbers in column (a) as found it before, which is numbers of 16 in column (a). Next, we will choose from column (a) a number that once we adding it, then the value of that number will have equality to the multiplier, which is 18 in this case. Then the numbers that we will choose are 2 and 16, since we know 2 + 16 = 18. We put this number in column (c) in the row like column (a) has. On column (d) we will put a number from column (b) that parallels with the numbers of 2 and 16 on column (a), which is 48 and 384. Moreover, we add the numbers that exist in column (d), which is 48 + 384, which gives us 432, which concludes our finding, about twenty-fourth of eighteen is four hundred and thirty-two, or 24 * 18 = 432. Of course, whoever can memorize the table of multiplication will do a “counting” of 24 * 18 is much more shorten time than people who performing this method of duplication.

**Division and/or Quotient**

Likely a multiplication is the other form of summation, then division is the other form of subtraction. If we divide between twelve and four, then we want to know about how many of fourth to perform 12. We will do several fundamental operations like:

12 – 4 = 8

8 – 4 = 4

4 – 4 = 0

With that information on how many times we need to perform, now we know to subtract 12 by 4, then we get 8, and subtracting 8 by 4, then we get 4, and subtracting 4 by 4, we can get 0. There is no extra number left, except 0. We now then define as 4 subtrahends around three times to get no extra number left to be subtrahend by 4. In this case, we know it now about 12 divide by 4, or 12: 4, is 3.

**Involution – Exponent**

Involution in Mathematics may not as famous as the other words that are mostly used, especially in the study of Mathematics operation. Involution is the other name of performing an Exponents. Which is exponent means we are giving a specific superscript to some main number, which is a number that given an exponent or a superscript. An example like if we want to give a superscript or an exponent of 3 to the number of 2, will be mean about we want to multiply that number of 2 around three times of similar that number itself, which is like 2 * 2 * 2. So, the numbers of 2 * 2 * 2, we can re-write it as 2^3, the main number of 2 with a small value that superscript is written on the right top of it, which called exponent of the main number. In this case like 2 to the power of 3, or 2 * 2 * 2, written as `2^3` .

Notes in here; 2 * 2 * 2 do not equal 2 * 3, even though in some simple cases they could be similar, like 2*2.

If the exponent is equal to 1, this means that numbers have not changed. So, 2 to the power of 1 will remains 2, or 3 to the power of 1, will remain 3, in geometrically written as `3^1 = 3` . If we are using the exponent of zero (0), we want to show that the main number itself wants to be divided by the main number itself. To whatever number with the exponent of zero (0) will always be equals to 1. An example like 4 to the power of 0 is one, `4^0 = 1` ,ten to the power of zero is one, `10^0 = 1` ,hundreds to the power of zero are one, `100^0 = 1` and so on.

There is also an exponent value of negative. A main number with the negative exponent will equal the division of that main number itself with the positive exponent, which in the form of what will be called a fraction, in short, the form will be like: ( 1 / main number ) with the related positive exponent. So, 5 to the power of minus one is 1 over 5 to the power of one, `5^-1 = 1/(5^1) = 1/ 5` ,three to the power of minus two is 1 over 3 to the power of two which also equals 1 over 9, `3^-2 = 1/(3^2) = 1/9` .

For more information about fractions, and negative numbers, please, refer to the 2nd Early Operational Tool in Arithmetic System.

**Evolution – Roots**

In the process of evolution system operation in mathematics, we will find about roots. From a basic number, we will find about the other form of that number, which if we multiplied by several times with those necessary numbers, will be a form of that initial number itself. A number that has been multiply with those numbers itself will be called roots. If we want to find out on which number that if we multiplied will give us the value of 4, been said about we are in the process of finding the roots of 4, which symbolically written as `sqrt(4)` . The number of 2, that we know from a multiplication, and if we multiplied with the number itself, in the form of factor 2, will giving a product of 4. Roots of the number of 4 can be also performed in negative numbers because the product of two negative numbers is a positive number, `sqrt(4) = 2` . The cube roots or root of the power of 3 from the number of 8, which written as `root(3)(8) = 2` ` ` , is a number that been used to factorized it three times, that can perform of 8, which is 2 * 2 * 2 = 8, therefore the cube roots of 8 are 2.

To be continued with information about fraction, negative number, the irrational number, an imaginary number.

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